Statistical Methods in Medical Research 4ed

586 Survival analysis
the death rates, from (17.25), are l0…t† for stage 3 and l0…t†exp…b† for stage 4, and exp…b†
is the death rate of stage 4 relative to stage 3. The first death occurred after 4 days (Table
17.5) when the risk set consisted of 19 stage 3 subjects and 61 stage 4 subjects. The death
was of a stage 4 subject and the probability that the one death known to occur at this time
was the particular stage 4 subject who did die is, from (17.27),
p1 ˆ exp…b†=‰19 ‡ 61 exp…b†Š:
The second time when deaths occurred was at 6 days. There were two deaths on this day
and this tie is handled approximately by assuming that they occurred simultaneously so
that the same risk set, 19 stage 3 and 60 stage 4 subjects, applied for each death. The
probability that a particular stage 3 subject died is 1=‰19 ‡ 60 exp…b†Š and that a particular
stage 4 subject died is exp…b†=‰19 ‡ 60 exp…b†Š and these two probabilities are
combined, using the multiplication rule, to give the probability that the two deaths consist
of the one subject from each stage,
p2 ˆ exp…b†=‰19 ‡ 60 exp…b†Š 2 :
Strictly this expression should contain a binomial factor of 2 (§3.6) but, since a constant
factor does not influence the estimation of b, it is convenient to omit it. Working through
Table 17.5, similar terms can be written down and the log-likelihood is equal to the sum of
the logarithms of the pj. Using a computer, the maximum likelihood estimate of b, b, is
obtained with its standard error:
b ˆ 0 9610,
SE…b† ˆ0 3856:
To test the hypothesis that b ˆ 0, that is, exp…b† ˆ1, we have the following as an
approximate standardized normal deviate:
Approximate 95% confidence limits for b are
z ˆ 0 9610=0 3856 ˆ 2 49 …P ˆ 0 013†:
0 9610 1 96 0 3856
ˆ 0 2052 and 1 7168:
Taking exponentials gives, as an estimate of the death rate of stage 4 relative to stage 3,
2 61 with 95% confidence limits of 1 23 and 5 57.
The estimate and the statistical significance of the relative death rate using
Cox's approach (Example 17.2) are similar to those obtained using the logrank
test (Example 17.1). The confidence interval is wider in accord with the earlier
remark that the confidence interval calculated using (17.16) has less than the
required coverage when the hazard ratio is not near to unity. In general,
when both the logrank test and Cox's proportional hazards regression model
are fitted to the same data, the score test (§14.2) from the regression approach
is identical to the logrank test (similar identities were noted in Chapter 15 in